Several characterizations of these majorization concepts exist. One characterization that is particularly useful in the context of optimization problems is in terms of objective values for Schur-convex functions. A function \(\Phi {:} \ \mathbb {R}^n \rightarrow \mathbb {R}\)  is said to be Schur-convex if it preserves the order of majorization, i.e., if \(\Phi (x) \le \Phi (y)\)  whenever \(x \prec y\) . The class of Schur-convex functions is broad and includes, as an important subclass, symmetric (quasi-)convex functions (see, e.g., Proposition 1.C.3 in [1]}). In particular, the following well-known characterizations of majorization and Schur-convex functions will be useful in the remainder of this article:
